Do BGE rigs have to be IK or FK?
They can be either! although FK rigs are much easier on the game engine's animation player, they're usually not as flexible (i.e. your player doesn't look where the camera does.) I tend to use a combo of both:

To start with, give the player animations (FK) then, simply add the constraint you want (the fewer the better) and add a "armature" actuator. Now this is automatically set to "run armature" but the setting can be changed to "enable/disable" (for individual constraints) or "set influence" or even "set target" which, although is resource consuming for your computer, can bring you some awesome mechanics that just aren't available for FK rigs.
And don't be fooled, not just the IK constraint can be used with the armature actuator, track-to and the whole kebab can be integrated into your game just as easily (with a few exceptions, I’ll look them up).

Now, usually, to get my player to look where I want, he needs a vertical head bone. I next add a cube at the location of the head bone, set it to no-collision, and add an empty above it. Parent cube to player, empty to cube, give mouse look logic to the cube and set the IK target of the head-bone to the empty and you're gold!